Mathematics Branches, Topics, and Sub-Topics

A structured visual guide to the major mathematical areas and their relationships.

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20Hxx Matrix groups

This subtopic studies matrix groups, including classical matrix groups and their algebraic and geometric properties.

Specific topics

20H05 Unimodular groups, congruence subgroups of matrix groups

Overview

20H05 studies unimodular groups, congruence subgroups of matrix groups in matrix groups. It studies groups realized as invertible matrices, emphasizing classical, arithmetic, and geometric subclasses.

Related Wikipedia Page

Matrix Groups (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for unimodular groups, congruence subgroups of matrix groups
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to connect linear actions with geometry, arithmetic, and subgroup structure.

Applications

  • Classical matrix groups
  • Arithmetic and geometric matrix groups
  • Structure and subgroup analysis

References

Recommended Textbooks

20H10 Fuchsian groups and their generalizations

Overview

20H10 studies fuchsian groups and their generalizations in matrix groups. It studies groups realized as invertible matrices, emphasizing classical, arithmetic, and geometric subclasses.

Related Wikipedia Page

Matrix Groups (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for fuchsian groups and their generalizations
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to connect linear actions with geometry, arithmetic, and subgroup structure.

Applications

  • Classical matrix groups
  • Arithmetic and geometric matrix groups
  • Structure and subgroup analysis

References

Recommended Textbooks

20H15 Other geometric groups, including crystallographic groups

Overview

20H15 studies other geometric groups, including crystallographic groups in matrix groups. It studies groups realized as invertible matrices, emphasizing classical, arithmetic, and geometric subclasses.

Related Wikipedia Page

Matrix Groups (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for other geometric groups, including crystallographic groups
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to connect linear actions with geometry, arithmetic, and subgroup structure.

Applications

  • Classical matrix groups
  • Arithmetic and geometric matrix groups
  • Structure and subgroup analysis

References

Recommended Textbooks

20H20 Other matrix groups over fields

Overview

20H20 studies other matrix groups over fields in matrix groups. It studies groups realized as invertible matrices, emphasizing classical, arithmetic, and geometric subclasses.

Related Wikipedia Page

Matrix Groups (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for other matrix groups over fields
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to connect linear actions with geometry, arithmetic, and subgroup structure.

Applications

  • Classical matrix groups
  • Arithmetic and geometric matrix groups
  • Structure and subgroup analysis

References

Recommended Textbooks

20H25 Other matrix groups over rings

Overview

20H25 studies other matrix groups over rings in matrix groups. It studies groups realized as invertible matrices, emphasizing classical, arithmetic, and geometric subclasses.

Related Wikipedia Page

Matrix Groups (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for other matrix groups over rings
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to connect linear actions with geometry, arithmetic, and subgroup structure.

Applications

  • Classical matrix groups
  • Arithmetic and geometric matrix groups
  • Structure and subgroup analysis

References

Recommended Textbooks

20H30 Linear groups over specific rings

Overview

20H30 studies linear groups over specific rings in matrix groups. It studies groups realized as invertible matrices, emphasizing classical, arithmetic, and geometric subclasses.

Related Wikipedia Page

Matrix Groups (Wikipedia)

Useful Links

Key Ideas

  • Canonical formulations and representative examples for linear groups over specific rings
  • How structural, local, and computational viewpoints interact in the subject
  • Standard theorem patterns and invariants used in current research practice

Typical Uses

Used to connect linear actions with geometry, arithmetic, and subgroup structure.

Applications

  • Classical matrix groups
  • Arithmetic and geometric matrix groups
  • Structure and subgroup analysis

References

Recommended Textbooks